texas calculatem key

88
Chapter 1
Functions and Linear Models
What to Use as xand y, and How to Interpret a Linear Model
m~/n a problem where I must find a linear relationship between two quantities, which
quentitydo I use as x and which dol use as y?
m:
The key is to decide which of the two quantities is the independent variable, and
which is the dependent variable. Then use the independent variable as x and. the
dependent variable as y. In other words, y depends on x.
Here are examples of phrases that convey this information, usually of the form Find
y {dependent variable} in terms'oi x [independent variable}:
• Find the cost in terms of the number
of items,y
• How does color depend on wavelength?
= C ost.x = ;; hems
y = Coror; X = -~·<.yeletig'[h
If no information is conveyed about which variable is intended to be independent,
then you can use whichever is convenient
El
= How do l interpret a general linear model y
m:
= mx + b?
The key to interpreting a linear mod~1 is to remember the units we use to measure
m and b:
The slope m is measured inlinitsof yper unit of x; the intercept bis measuredin units ofy.
For instance, if y = 4.3x + 8.1 and you know that xis measured in feet and y in
kilograms, then you can already say, "y is 8.1 kilograms when x = o feet, and increases at a rate of 4.3 kilograms per foot" without even knowing anything more
about the situation!
1.3 EXERCISES
v more advanced
[il indicates
•
challenging
exercises that should be solved using technology
In Exercises J-6, a table of values for a linear function is given.
Fill in the missing value and calculate m in each case.
1,=.
-1
0
5,~
~
6,~
1
~
~
2.~
=
4,=
3,~
In Exercises 7-JO,jirstjindf(O),
7'1
~
~
if not supplied, and then find
the equation a/the given linear/unction.
8'1
x
-2
0
2
4
f(x)
-1
-2
-3
-4
x
-6
-3
0
3
f(,:r:)
1
2
3
4
1.3 Linear Functions and Models
x
-3
-4
-1
I(x)
4
f(x)
-2
-1
-2
-3
-4
2
3
4
6
8
10
=9
36.3y=4
37.2x
= 3y
38. 3x =-2y
In Exercises 39-54, calculate the slope, if defined, of the straight
line through the given pair of points. fly to do as many as you can
without writing anything down except the answel: HINT [See Quick
Examples on page 79.]
-Jp each of Exercises J J - J 4, decide which of the two given
.f...1!ctipnsis linear and jind its equation. H"lT [See Example 1.]
I
x
0
1
2
3
4
~
f(x)
6
10
14
18
22
8
10
12
16
22
I g(x)
35.6y
89
39. (0, 0) and (1, 2)
40. (0, 0) and ( -1, 2)
41. (-I,
42. (2, 1) and (0, 0)
-2) and (0, 0)
43. (4, 3) and (5,1)
44. (4,3) and (4,1)
45. (1, -1) and (1, -2)
46. (-2,2)
47. (2, 3.5) and (4,6.5)
48. (10, -3.5) and (0, -1.5)
and(-I,
-1)
49. (300, 20.2) and (400, 11.2)
50. (1, -20.2)
and (2, 3.2)
t
x
-10
0
10
20
30
•
f(x)
-1.5
0
l.5
2.5
3.5
51.(0,1)
f
g(x)
-9
-4
1
6
11
52. (~,
x
0
3
6
10
15
54. (a, b) and (e, b) (a :f. c)
1
i
f(x)
0
3
5
7
9
55. In the following figure, estimate the slopes of all line segments.
I
g(x)
-1
5
11
19
29
l
I
and
1) and
(-1'~)
(-~,~)
53. (a, b) and (e, d) (a :f. c)
j
I
--1----
i
I
I
x
0
3
5
6
9
!
!I
f(X}
2
6
9
12
IS
I I
l
g(x)
-\
8
14
17
26
!
15-24, find
the slope of the given line,
if
'i~"lned.
2r:
x+
i~.3x
1
2x -1
18.y=---
6
3
20. 8x -2y = 1
+ 1= 0
!1.3y+ 1 = 0
~'-+x
+3y
I
56. In the following figure, estimate the slopes of all line segments.
16.y= -+4
3
1.\'= --
-
(f)j
1.--.....L..----''----'------'-__ -'----'-_.L........J
3
1~'Y=-2x-4
_
it is
i
i
I!----''\.c:! -'---'---\
I
:'! Exercises
(a)1
+_-+_~...L-_/
'---1-
=7
22. 2x +3
=0
24. 2y+3
=0
--:::xercises 25.,.38, graph the given equation. HINT [See Quick
::,zcples on page 77.]
:5.y
= 2x -
:-.y
= -~x +2
.-
I
-"·Y+4x=-
1
4
2y = 7
:1. ix
-
:3. 3x
=8
26.y
=X
28.y
=
-
3
-1x +3
30.y- ~x =-2
32. 2x - 3y
34.2x =-7
=
1
In Exercises 57-74,jind a linear equation whose graph is the
straight line with the given properties. HINT [See Example 2.]
57. Through (1, 3) with slope 3
58. Through (2, 1) with slope 2
90
Chapter 1
59. Through (1,
Functionsand Linear Models
-i) with slope
60. Through (0, -~)
79. Demand Sales figures show that your company sold 1,960
pen sets each week when they were priced at $l/pen set
and 1,800 pen sets each week when they were priced at
$5/pen set. What is the linear demand function for your
pen sets? HUn [See Example 4.]
~
with slope ~
61. Through (20, -3.5)
per unit of x.
and increasing at a rate of 10 units ofy
62. Through (3.5, -10)
per 2 units of x.
and increasing at a rate of I unit of y
80. Demand A large department store is prepared to buy 3,950
of your neon-colored shower curtains per month for $5 each,
but only 3,700 shower curtains per month for $10 each. What
is the linear demand function for your neon-colored shower
curtains?
63. Through (2, -4) and (1, 1)
64. Through(l,
-4) and(-I,
65. Through (I, -0.75)
66. Through (0.5, -0.75)
-I)
81. Demand for Cell Phones The following table shows worldwide sales of Nokia cell phones and their average wholesale
prices in 2004Y
and (0.5, 0.75)
and (1, -3.75)
67. Through (6,6) and parallel to the line x
+y
Quarter
=4
Wholesale Price ($)
68. Through (1/3, -1) and parallel to the line 3x - 4Y = 8
Sales (millions)
Second
Fourtb
III
45.4
105
51.4
69. Through (0.5, 5) and parallel to the line 4x - 2y = 11
70. Through (1/3, 0) and parallel to the line 6x - 2y
a. Use the data to obtain a linear demand function for
(Nokia) cell phones, and use your demand equation to
predict sales if Nokia lowered the price further to $103.
b. Fill in the blanks: For every __
increase in price, sales
of cell phones decrease by _
units.
11
Through (0, 0) and (p, q)
7l.i
72. ,;-Through (p, q) and parallel to y = rx
73. : Through (p, q) and (r, q) (p
74.
=
1/
Through (p, q) and (r, s)(p
+s
¥ r)
82. Demand for Cell Phones The following table shows projected worldwide sales of (aU) cell phones and wholesale
prices:28
i= r)
APPLICATIONS
75. Cost The RideEm Bicycles factory can produce 100 bicycles in a day at a total cost of$10,500 and it can produce 120
bicycles in a day at a total cost of $Il,OOO. What are the
company's daily fixed costs, and what is the marginal cost
per bicycle? HIMT [See Example 3.]
76. Cost A
of soda
of soda
weekly
Wholesale Price ($)
100
80
600
800
83. Demand for Monorail Service, Las Jlegas In 2005, the LaS
Vegas monorail charged $3 per ride and had an average
ridership of about 28,000 per day. In December, 2005
the Las Vegas Monorail Company raised the fare to $5 per
ride, and average ridership in 2006 plunged to around
19,000 per day.29
a. Use the given information to find a linear demand
equation.
b. Give the units of measurement and interpretation of
the slope.
c. What would be the effect on ridership of raising the fare
to $6 per ride?
100 iPods.
78. Cost: Xboxes If it costs Microsoft $4,500 to manufacture 8
Xbox 360s and $8,900 to manufacture 16,t obtain the corresponding linear cost function. What was the cost to
manufacture each additional Xbox? Use the cost function
to estimate the cost of manufacturing 50 Xboxes.
27
2007, Volume 14, No. 14, www.manufacturingnews.com.
tSource for estimate of marginal cost: www.isuppli.com.
2008
a. Use the data to obtain a linear demand function for cell
phones, and use your demand equation to predict sales if
the price was set at $85.
b. Fill in the blanks: For every __
increase in price, sales
of cell phones decrease by _
units.
77. Cost: iPods It cost Apple approximately $800 to manufacture 5 30-gigabyte video iPods and $3,700 to manufacture 25. * Obtain the corresponding linear cost function.
What was the cost to manufacture each additional iPod?
Use the cost function to estimate the cost of manufacturing
& Technology News, July 31,
2004
Sales (millions)
soft-drink manufacturer can produce 1,000 cases
in a week at a total cost of $6,000, and 1,500 cases
at a total cost of $8,500. Find the manufacturer's
fixed costs and marginal cost per case of soda.
"Source for cost data: Manufacturing
Year
Source: Embedded.comlCompanyreports December 2004.
Wholesale price projections are the authors'. Source for sales
prediction: I-StatINDR December, 2004.
29Source:New York TImes, February 10, 2007, p. A9.
28
1.3 Linear Functionsand Models
a. Use these data to express h, the annual Martian imports of
mercury (in millions of kilograms ), as a linear function of t,
the number of years since 2010.
b. Use your model to estimate Martian mercury imports in
2230, assuming the import trend continued.
84. Demand for Monorail
Service,
Mars The Utarek
monorail, which links Jhe three urbynes (or districts) of
Utarek, Mars, chargedZ5 per ride30 and sold about 14 million rides p~ day. When the Utarek City Council lowered
the fare toZ3 per ride, the number of rides increased to
18 million per day.
89. Satellite Radio Subscriptions The number of Sirius Satellite
Radio subscribers grew from 0.3 million in 2003 to
3.2 million in 2005.32
a. Use the given information to find a linear demand
equation.
b. Give the units of measurement and interpretation of the
slope.
c. What w.Quldbe the effect on ridership of raising the
fare toZ 10 per ride?
a. Use this information to find a linear model for the
number N of subscribers (in millions) as a function of
time t in years since 2000.
b. Give the units of measurement and interpretation of the
slope.
c. Use the model from part (a) to predict the 2006 figure.
(The actual 2006 figure was approximately 6 million
subscribers.)
85. Equilibrium Price You can sell 90 pet chias per week if they
are marked at $1 each, but only 30 each week if they are
marked at $2/chia. Your chia supplier is prepared to sell you
20 chias each week if they are marked at $l/chia, and 100
each week if they are marked at $2 per chiao
90. Freon Production The production of ozone-layer damaging Freon 22 (chlorodifluoromethane)
in developing
countries rose from 200 tons in 2004 to a projected
590 tons in 2010.33
a. Write down the associated linear demand and supply
functions.
b. At what price should the chias be marked so that there is
neither a surplus nor a shortage of chias? HINT [See
a. Use this information to find a linear model for the
Example 4.]
amount F of Freon 22 (in tons) as a function of time t in
years since 2000.
b. Give the units of measurement and interpretation of the
slope.
c. Use the model from part (a) to estimate the 2008 figure
and compare it with the actual projection of 400 tons.
86. Equilibrium Price The demand for your college newspaper is
2,000 copies each week if the paper is given away free of
charge, and drops to 1,000 each week if the charge is
lO¢/copy. However, the university is prepared to supply only
600 copies per week free of charge, but will supply 1,400
each week at 20¢ per copy.
91. Velocity The position of a model train, in feet along a
railroad track, is given by
a. Write down the associated linear demand and supply
s(t) = 2.5t
functions.
b. At what price should the college newspapers be sold
so that there is neither a surplus nor a shortage of
papers?
+ 10
after t seconds.
a. How fast is the train moving?
b. Where is the train after 4 seconds?
c. When will it be 25 feet along the track?
87. Pasta Imports During the period 1990-2001, U.S. imports
of pasta increased from 290 million pounds in 1990 (t
by an average of 40 million pounds/year."
91
= 0)
92. Velocity The height of a falling sheet of paper, in feet from
the ground, is given by
s(t) = -1.8t
a. Use these data to express q, the annual U.S. imports of
pasta (in millions of pounds), as a linear function of t,
the number of years since 1990.
b. Use your model to estimate U.S. pasta imports in 2005,
assuming the import trend continued.
+9
after t seconds.
a. What is the velocity of the sheet of paper?
b. How high is it after 4 seconds?
c. When will it reach the ground?
93. ~ Fast Cars A police car was traveling down Ocean Parkway
in a high speed chase from Jones Beach. It was at Jones
Beach at exactly 10 pm (t = 10) and was at Oak Beach, 13
miles from Jones Beach, at exactly 10:06 pm.
88. Mercury Imports During the period 2210-2220, Martian
imports of mercury (from the planet of that name) increased
from 550 million kg in 2210 (t = 0) by an average of
60 million kg/year.
a. How fast was the police car traveling? HINT [See Example 6.]
b. How far was the police car from Jones Beach at time t?
: Z designates Zonars, the official currency in Mars. See www.marsnext
.:m for details of the Mars colony, its commerce, and its culture.
32
Jam are rounded. Sources: Department of Commerce/New York Times,
o=:'tember5, 1995, p. D4; International Trade Administration, March
~'. 2002, www.ita.doc.gov/.
JJ
Figures are approximate. Source: Sirius Satellite Radio/New York
TImes, February 20, 2008, p. AI.
Figures are approximate. Source: Lampert Kuijpers (Panel of the
Montreal Protocol), National Bureau of Statistics in China, via CEIC
DSataiNew York TImes, February 23, 2007, p. CL
1.4
Linear Regression
101
Substituting these values into the formula we get
n{L:xy) - (L:x)(L:y)
r=-r============~~============7
jn(L:x2)
- (L:x)2. jn(L:y2)
- (L:y)2
8(670) - (56)(88)
-/8(560) - 562
.
-/8(986) - 882
~ 0.982.
Thus, the fit is a fairly good one, that is, the original points lie nearly along a straight
line, as can be confirmed from the graph in Example 2.
EXERCISES
,T
-nore advanced
•
challenging
·Ii indicates exercises that should be solved using technology
,;i1zE-r:ercises 1-4, compute the sum-of-squares error (SSE) by hand
).,'fir1r the given set of data and linear model HINT [See Example 1.]
1. (1,1), (2, 2), (3, 4);
y = x-I
2. (0,1), (1, 1), (2, 2);
y
n
14. a. {(I, 3), (-2, 9), (2, I)}
b. {(O, 1), (1, 0), (2, I)}
c. {CO,0), (5, -5), (2, -2.1)}
APPLICATIONS
=x +I
=
-x
15. Worldwide Cell Phone Sales Following are forecasts of worldwide annual cell phone handset sales."
+2
(0, -1), (I, 3), (4, 6), (5, 0);
y
(2,4), (6, 8), (8, 12), (10, 0);
y = 2x - 8
Year x
Exercises 5-8, use technology to compute the sum-of
squares error (SSE) for the given set of data and linear models.
indicate which linear model gives the better fit.
III
= 1.5x - 1 b.y = 2x - 1.5
6. (0,1), (1, 1), (2, 2); a.y = OAx + 1.1 b.y = 0.5x + 0.9
7. (0, -1), (I, 3), (4, 6), (5, 0); a.y = O.3x + l.l
b.y = OAx + 0.9
(1,1), (2, 2), (3, 4);
a.y
8. (2, 4), (6,8), (8,12), (l0, 0);
a.y
b.y
=
-O.lx
+7
(3, 4)
10. (0,1), (1,1), (2, 2)
n, (0,
=
(x
3 represents 2003). Complete the following table and
obtain the associated regression line. (Round coefficients to 2
decimal places.) HINT [See Example 3.]
= -0.2x + 6
Find the regression line associated with each set of points in
Exercises 9-12. Graph the data and the best-fit line. (Round all
zoefficients to 4 decimal places.) HINT [See Example 2.]
9. (1, 1),(2,2),
Salesy (millions)
-1), (1, 3), (4, 6), (5, 0)
x
y
3
500
5
600
7
800
xy
Xl
I Totals
Use your regression equation to project the 2008 sales.
16. Investment ill Gold Following are approximate values of the
Amex Gold BUGS Index."
U. (2,4), (6, 8), (8,12), (10, 0)
ln the next two exercises, use correlation coefficients to deter-nine which of the given sets of data is best fit by its associated
regression line and which is fit worst. Is it a perfect fit for any of
:;je data sets? HINT [See Example 4.]
45Source: In-StatMDR,
13. a. {(l, 3), (2, 4), (5, 6)}
b. {CO,-1), (2,1), (3, 4)}
c. {(4, -3),
(5, 5), (0, O)}
www.in-stat.conv.
July, 2004.
46BUGS stands for "basket of unhedged gold stocks." Figures are approximate. Sources: www.321gold.com. Bloomberg Financial Markets!
New York Times, Sept 7, 2003, p. BU8, www.amex.com.
102
Chapter 1
Functions and Linear Models
(x = 0 represents 2000). Complete the following table and
obtain the associated regression line. (Round coefficients to
2 decimal places.)
xy
20. Oil Recovery (Refer to Exercise 19.) The following table
gives the approximate economic value associated with various levels of oil recovery in Texas. so
Percent Recovery (%)
xl
x
y
0
50
4
250
Find the regression line and use it to estimate the economic
value associated with a recovery level of 70%.
7
470
21.riB Soybean Production The following table shows soybean
I Totals
Economic Value ($ billions)
production, in millions of tons, in Brazil's Cerrados region, as
a function of the cultivated area, in millions of acres. 51
Use your regression equation to estimate the 2005 index to
the nearest whole number.
Area (millions of acres)
17. E-Commerce The following chart shows second quarter total
retail e-cornmerce sales in the United States in 2000, 2004,
and 2007 (t = 0 represents 2000):47
Production
(millions oftons)
25
30
32
40
52
15
25
30
40
60
a. Use technology to obtain the regression line, and to show
a plot of the points together with the regression line.
(Round coefficients to two decimal places.)
b. Interpret the slope of the regression line.
Yeart
Sales ($ billion)
Find the regression line (round coefficients to two decimal
places) and use it to estimate second quarter retail e-commerce
sales in 2006.
table shows u.s.
exports to Taiwan as a function of U.S. imports from Taiwan,
based on trade figures in the period 1990-2003.52
22. !IT Trade with Taiwan The following
Imports
18. Retail Inventories The following chart shows total January
retail inventories in the United States in 2000, 2005, and 2007
(t = 0 represents 2000):48
a. Use technology to obtain the regression line, and to show
a plot of the points together with the regression line.
(Round coefficients to two decimal places.)
b. Interpret the slope of the regression line.
Yeart
Inventory
($ billions)
Exports ($ billions)
($ billion)
Find the regression line (round coefficients to two decimal
places) and use it to estimate January retail inventories in 2004.
~ Exercises 23 and 24 are based on theJollowing table comparing the number of registered automobiles, trucks, and motorcycles in Mexico Jor various years from 1980 to 2005.53
19. Oil Recovery The Texas Bureau of Economic Geology
published a study on the economic impact of using carbon
dioxide enhanced oil recovery (EOR) technology to extract
additional oil from fields that have reached the end of their
conventional economic life. The following table gives the approximate number of jobs for the citizens of Texas that would
be created at various levels of recovery."
Year
Automobiles
(millions)
Trucks
(millions)
Motorcycles
(millions)
1980
4.0
1.5
0.28
1985
5.3
2.1
0.25
1990
6.6
3.0
0.25
1995
7.5
3.6
0.13
2000
10.2
4.9
0.29
2005
14.7
7.1
0.61
Percent Recovery (%)
Jobs Created (millions)
Find the regression line and use it to estimate the number of
jobs that would be created at a recovery level of50%.
50Ibid.
47Figures are rounded. Source: US Census Bureau, www.census.gov,
November 2007.
4glbid.
49Source:"C02-Enhanced Oil RecoveryResource Potentialin Texas:
Potential PositiveEconomic Impacts;' TexasBureau of Economic
Geology,April 2004, www.rrc.state.tx.usltepclC02-EOR_whitc....paper.pdf.
51 Source: Brazil Agriculture Ministry/New York Times, December 12,
2004, p. N32.
52 Source: Taiwan Directorate General of Customs/New
York Times,
December 13,2004, p. C7.
53
Source: Instituto Nacional de Estadistica y Geografia (lNEGI),
www.inegi.org.mx.